PLL Circuit and Method for Generating a Modulated Carrier Signal

CROSS-REFERENCE TO RELATED APPLICATIONS

The present disclosure is based on PCT filing PCT/EP2023/050324, filed Jan. 9, 2023, which claims priority from European Patent Application No. 22154628.6, filed Feb. 1, 2022, the entire contents of each are incorporated herein by reference. FIELD The present disclosure generally relates to phase locked loop (PLL) circuits, and, more particularly, to circuits and methods for compensating phase errors in two-point modulated PLLs.

BACKGROUND

An increasing demand for wideband mobile wireless services mandates new architectures and solutions for low-cost radio-frequency transceivers. Digital polar transmitters (TXs) entail an advantage of eliminating power-consuming blocks as digital-to-analog converters (DACs), filters and mixers typical of conventional quadrature transmitters and of allowing the use of high-efficiency digitally controlled power amplifiers. A two-point signal-injection scheme (also known as two-point modulation) may overcome bandwidth limitations of PLLs and allow a wideband frequency or phase modulation of frequency synthesizers. In in two-point modulated PLLs, the modulation signal (for example, phase or frequency) may be sent to both a controlled oscillator's modulation input as well as a PLL master oscillator. Examples of a controlled oscillator include Voltage Controlled Oscillators (VCOs), Digitally Controlled Oscillators (DCOs) or Numerically Controlled Oscillators (NCOs). An impairment arising in two-point modulation schemes at modulation bandwidths is represented by a delay spread between the two points of injection. This may significantly degrade the error vector magnitude (EVM) and may increase the out-of-band emission. Thus, it is an object of the present disclosure to provide methods and circuits for improved EVM in two-point modulated PLLs.

SUMMARY

According to a first aspect of the present disclosure, it is provided a PLL circuit for generating a modulated carrier signal. The PLL circuit comprises a controlled oscillator configured to generate a controlled oscillator output signal as the modulated carrier signal. A first input of the PLL circuit is configured to receive a desired phase change as a modulation signal at an n-th cycle of a non-uniform clock derived from the controlled oscillator output signal and a uniform reference clock. The PLL circuit comprises a first path (direct modulation path) for the desired phase change to the controlled oscillator to change its oscillation frequency based on the desired phase change. A second input of the PLL circuit is configured to receive a frequency control word (FCW) at the n-th cycle of the non-uniform clock. The FCW denotes a ratio of a desired carrier frequency of the modulated carrier signal and a frequency of the uniform reference clock. The PLL circuit comprises a second path (phase prediction path) for the desired phase change to a frequency control word accumulator configured to accumulate the desired phase change and the FCW to generate a first phase prediction of the controlled oscillator's output phase for the (n+1)-th cycle of the non-uniform clock. The PLL circuit comprises a non-uniform clock compensation circuit configured to recursively calculate, using the desired phase change, the frequency control word FCW and the first phase prediction, a non-uniform clock compensation value for the desired phase change on the first path and/or a second phase prediction of the controlled oscillator's output phase for the (n+1)-th cycle of the uniform reference clock. A phase detector of the PLL circuit is configured to estimate an error between the second phase prediction and an instantaneous time offset between the uniform reference clock and the controlled oscillator output signal and to generate a control signal for the controlled oscillator based on the estimated error. According to a second aspect, it is provided a method for generating a modulated carrier signal. The method includes providing a controlled oscillator configured to generate a controlled oscillator output signal as the modulated carrier signal. The method includes receiving a desired phase change as a modulation signal at an n-th cycle of a non-uniform clock derived from the controlled oscillator output signal and a uniform reference clock. The desired phase change is provided to the controlled oscillator to change its oscillation frequency based on the desired phase change. The method includes receiving a frequency control word (FCW) at the n-th cycle of the non-uniform clock, wherein the FCW is a ratio of a desired carrier frequency of the modulated carrier signal and a frequency of the uniform reference clock. The method includes providing FCW and the desired phase change to a frequency control word accumulator configured to accumulate the desired phase change and the FCW to generate a first phase prediction of the controlled oscillator's output phase for the (n+1)-th cycle of the non-uniform clock. The method includes recursively calculating, using the desired phase change, the FCW, and the first phase prediction, a non-uniform clock compensation value for the desired phase change on the first path and/or a second phase prediction of the controlled oscillator's output phase for the (n+1)-th cycle of the uniform reference clock. The method includes estimating an error between the second phase prediction and an instantaneous time offset between the uniform reference clock and the controlled oscillator output signal and generating a control signal for the controlled oscillator based on the estimated error. The present disclosure presents a concept to compensate phase errors in two-point modulated PLLs, caused by a non-uniform digital processing clock. For the direct modulation path, it may correct phase modulation errors due to a time-varying phase accumulation time in the controlled oscillator. For the phase prediction path, it may correct a phase prediction error due to a variable offset between the uniform reference clock and the non-uniform digital processing clock. Finally, the present disclosure may help the PLL-based phase modulator to break the EVM barrier around 1 ⁢ 20 ⁢ log 10 ( 1 F ⁢ C ⁢ W ) . BRIEF DESCRIPTION OF THE FIGURES Some examples of apparatuses and/or methods will be described in the following by way of example only, and with reference to the accompanying figures, in which FIG. 1 shows an example of a PLL system where digital part is clocked by FREF; FIG. 2 shows a timing-diagram of the PLL-based phase modulator shown in FIG. 1 ; FIG. 3 shows an example of a PLL system where digital part is clocked by CKR 2 ; FIG. 3 a shows an example circuit of generating CKR 2 based on FREF and CKV; FIG. 4 shows a timing diagram of the PLL-based phase modulator shown in FIG. 3 ; FIG. 5 shows a PLL-based phase modulator with the proposed Non-Uniform Clock Compensation (NUCC); FIG. 6 shows NUCC details for the n-th FREF/CKR 2 clock cycle; and FIGS. 7 , 8 show PLL-based phase modulators with NUCC and DCO nonlinearity digital pre-distortion.

DETAILED DESCRIPTION

Some examples are now described in more detail with reference to the enclosed figures. However, other possible examples are not limited to the features of these embodiments described in detail. Other examples may include modifications of the features as well as equivalents and alternatives to the features. Furthermore, the terminology used herein to describe certain examples should not be restrictive of further possible examples. Throughout the description of the figures same or similar reference numerals refer to same or similar elements and/or features, which may be identical or implemented in a modified form while providing the same or a similar function. The thickness of lines, layers and/or areas in the figures may also be exaggerated for clarification. When two elements A and B are combined using an “or”, this is to be understood as disclosing all possible combinations, i.e., only A, only B as well as A and B, unless expressly defined otherwise in the individual case. As an alternative wording for the same combinations, “at least one of A and B” or “A and/or B” may be used. This applies equivalently to combinations of more than two elements. If a singular form, such as “a”, “an” and “the” is used and the use of only a single element is not defined as mandatory either explicitly or implicitly, further examples may also use several elements to implement the same function. If a function is described below as implemented using multiple elements, further examples may implement the same function using a single element or a single processing entity. It is further understood that the terms “include”, “including”, “comprise” and/or “comprising”, when used, describe the presence of the specified features, integers, steps, operations, processes, elements, components and/or a group thereof, but do not exclude the presence or addition of one or more other features, integers, steps, operations, processes, elements, components and/or a group thereof. Two-point modulation may be used in PLL-based phase or frequency modulators to avoid that the PLL tracks the phase modulation (PM) or frequency modulation (FM). An example for such a system and its timing diagram are shown in FIG. 1 and FIG. 2 , respectively. FIG. 1 shows an example of a phase modulated PLL system 100 . PLL system 100 comprises a first input 102 configured to receive, at each clock cycle of a uniform reference clock FREF, a desired phase ϕ M (all the phases shown in this disclosure may be normalized by ½π). The desired phase OM may be regarded as modulation signal for modulating an output signal CKV of a DCO (Digital Controlled Oscillator) 110 . The uniform reference clock FREF may be a highly stable clock signal derived from a crystal oscillator, for example. In the illustrated example, the desired phase PM is differentiated by an optional differentiator 104 which is coupled to first input 102 to obtain a desired phase change Δϕ M during each clock cycle of the uniform reference clock FREF. The skilled person having benefit from the present disclosure will appreciate that the desired phase change Δϕ PM could also directly be fed to first input 102 , for example, in case of frequency modulation (FM). The desired phase change Δϕ M may be applied to the PLL system 100 through two paths. The target of this so called two-point modulation is to cancel the PM within the PLL loop, since the PLL ideally only should track and stabilize a carrier phase. That is, PLL control signal Δϕ T provided by the PLL's loop filter 106 should only reflect an error in f c0 denoting an instantaneous DCO frequency and correct it to maintain a pure desired carrier frequency f C . The illustrated PLL system 100 comprises a first path 108 (direct modulation path) for the desired phase change Δϕ M from first input or differentiator 104 to DCO 110 in order to change the oscillation frequency of DCO 110 based on the desired phase change Δϕ M . The skilled person having benefit from the present disclosure will appreciate that DCO 110 could also be implemented as a VCO or NCO, for example. The first path 108 , the DM-path (direct modulation path), is fed to the DCO 110 . In response to the desired phase change Δϕ M , the DCO 110 changes its oscillation frequency by Δf M and accumulates a phase change Δϕ M in an output phase ϕ V of the DCO ϕ V ( t ) = ∫ 0 t f V ( t ) ⁢ d ⁢ τ = ∫ 0 t [ f C + Δ ⁢ f M ( t ) ] ⁢ d ⁢ τ . Equation ⁢ 1 The skilled person having benefit from the present disclosure will appreciate that Equation 1 is related to an ideal DCO without any phase noise. That is, only intended signals are present. PLL system 100 comprises a second input 112 configured to receive a frequency control word (FCW) during each clock cycle of the uniform reference clock FREF of PLL system 100 . FCW is a ratio of the desired carrier frequency f C of the carrier signal and a frequency f REF of the uniform reference clock. By ignoring an analog circuit delay between Δf M settling and Δϕ M updating ( i . e . Δ ⁢ f M [ t ] = Δϕ M [ t ] T R ⁢ E ⁢ F ) , Equation 1 can be simplified to ϕ V ( t ) = ∫ 0 t F ⁢ CW + Δϕ M ( t ) T R ⁢ E ⁢ F ⁢ d ⁢ τ , where ⁢ FCW = f c F R ⁢ E ⁢ F Equation ⁢ 2 (Frequency Control Word) is the ratio of f C and the digital PLL reference clock rate FREF (f REF ), and T R ⁢ E ⁢ F = 1 f R ⁢ E ⁢ F is the period of FREF. In the example shown in FIG. 1 and FIG. 2 , the synthesized digital clock is FREF. Therefore, at the n-th falling edge of FREF, Equation 2 reaches the expected phase value ϕ V [ n ] = ∑ i = 1 n - 1 { Δ ⁢ ϕ M [ i ] } + ( n - 1 ) · FCW = ϕ R [ n - 1 ] . Equation ⁢ 3 A second path 114 for the desired phase change Δϕ M , the PP-path (phase prediction path), may be fed to a FCW accumulator 116 , which may accumulate FCW and Δϕ M . The accumulated phase ϕ R [n]=Σ i=1 n {Δϕ M [i]+FCW}=Σ i=1 n {Δϕ M [i]}+n·FCW for the n-th FREF cycle ideally should predict the DCO's output phase at the beginning of the next (n+1) FREF cycle (ϕ V [n+1]). Thus, the FCW accumulator 116 is configured to accumulate the desired phase change Δϕ M and the FCW to generate a phase prediction ϕ R [n] of the DCO's output phase at the beginning of the next (n+1) FREF cycle. In FIG. 2 , it can be seen that of ϕ′ V [n] always reaches the value ϕ′ R [n−1] at the n-th FREF edge, i.e., the predicted phase ϕ′ R and accumulated DCO output phase ϕ′ V of are matching. The extra primes on the variables indicate that the influence of the carrier f C has been excluded compared to their original counterparts, i.e., ϕ V ′ ( t ) = ϕ V - f C · t = ∫ 0 t Δϕ M ( τ ) T R ⁢ E ⁢ F ⁢ d ⁢ τ ⁢ and ϕ R ′ [ n ] = ϕ R [ n ] - n · FCW = ∑ i = 1 n ⁢ { Δ ⁢ ϕ M [ i ] } . As can be seen from FIG. 1 , phase prediction ϕ R may be split into an integer and a fractional part. The integer part ϕ R,int may only be used during initial frequency locking phase to lock-on to the desired carrier frequency f C . The fractional part ϕ R,frac of phase prediction ϕ R may be used to track the carrier phase ϕ V when the PLL operates in a locked state. This split in ϕ R,int and ϕ R,frac is optional, but is usually beneficial for PLL architectures, since the integer part can be switched off, as soon as the PLL has achieved frequency lock. The fractional part ϕ R,frac is coupled to a phase detector 118 of PLL circuit 100 which is configured to estimate a phase error by comparing the expected/predicted fractional phase ϕ R,frac with an instantaneous time offset Δt S between the uniform reference clock FREF and DCO output signal CKV or signal CKR derived thereof, as can be seen in the timing diagram in FIG. 2 . The time offset Δt S can be measured between a falling edge of FREF and the next falling edge of CKV or CKR (which is synchronized with CKV but may have the same average frequency as FREF). A signal (1−ϕ R,frac )·T CKV [n] (with T CKV [n] denoting the period of CKV during the n-th clock cycle) may be compared to Δt S in order to obtain the error signal (phase error) Δt e [n]=(1−ϕ R,frac [n−1])·T CKV [n]−Δt S [n]. The error signal Δt e [n] (of phase detector 118 may be provided to loop filter 106 via time-to-digital converter (TDC) 126 . An output signal Δϕ T of loop filter 106 is then fed back as control signal to DCO 110 . In accordance with the principle of PLLs, if only FCW without any PM is fed to the DCO, DCO 110 generates a periodic DCO output signal CKV, and the phase detector 118 compares the phase ϕ V of that signal with the phase prediction ϕ R , adjusting the DCO 110 to keep the phases matched. If some PM is only present in DCO output signal CKV through the DM-path 108 , this would cause some “virtual phase error” in the phase detection. However, since the same PM is also fed to the PLL via the PP-path 114 , the PM is ideally fully cancelled, i.e., only a carrier phase error is tracked within the PLL. So far, the PLL-based phase modulator discussion was based on an ideal assumption that the synthesized digital part works with a uniform clock FREF. Here, uniform clock refers to a clock where each clock cycle has the same duration without jitter. However, in practice this assumption of uniform clock is not true, since most digital intensive PLLs preferably use a non-uniform digital reference clock which is phase-aligned with the DCO output signal CKV. Non-uniform clock refers to a clock where different clock cycles may have different durations. A non-uniform digital reference clock which is phase-aligned with the DCO output signal CKV benefits the system in three aspects. First, it may help to minimize the disturbance of the DCO 110 while updating its Δf T and Δf M . The main sources of DCO disturbance in case the uniform reference clock FREF is not phase aligned with CKV are due increased instantaneous phase errors at the update time of the DCO inputs (which then are integrated on the DCO output). Secondly, it may protect Δt S sampling in phase detector 118 from disturbance due to updating Δf M . As shown in FIG. 2 , if FREF is used as the digital clock, Δf M is updated at the exact beginning of Δt S sampling. However, due to analog delay, Δf M update cannot propagate instantaneously. This distorts the sampled Δt S , and finally introduces undesired errors in the PLL loop. Third, it may avoid metastability issues in certain digital blocks, for instance sampler 120 in FIG. 1 , which is clocked by uniform reference clock FREF but samples an output of counter/accumulator 122 from non-uniform CKV clock domain. FIG. 3 shows another example of a PLL system 300 where first input 102 and second input 112 are clocked by a non-uniform clock CKR 2 which may be generated by snapshot circuit 124 based on the DCO output signal CKV and the uniform reference clock FREF. In contrast, in PLL system 100 of FIG. 1 , the first input 102 and second input 112 are both clocked by uniform reference clock FREF. An example of snapshot circuit 124 is depicted in FIG. 3 a . The non-uniform clock CKR 2 may be generated based on the DCO output signal CKV and the uniform reference clock FREF using a logic circuit comprising various gates and flipflops implementing a frequency divider circuit. As can be seen, the DCO output signal CKV and the uniform reference clock FREF have different respective frequencies and are asynchronous with respect to each other. A frequency of the DCO output signal CKV essentially corresponds to the phase modulated carrier frequency, while a frequency of the uniform reference clock FREF may be substantially lower and highly stable. The non-uniform clock CKR 2 and the uniform reference clock FREF have equal respective average frequencies, however, instantaneous periods and start points of the non-uniform clock CKR 2 relative to the uniform reference clock FREF do vary from cycle to cycle. In the illustrated example, a rising edge of non-uniform clock CKR 2 is phase-aligned with the second falling edge of CKV after the falling edge of FREF. The skilled person having benefit from the present disclosure will appreciate that falling edges may also be rising edges and vice versa in different embodiments. As described above, the digital non-uniform clock CKR 2 may be phase-aligned with the second CKV falling edge after the falling FREF edge (see FIG. 3 a , 4 ). CKR 2 may be generated by snapshot circuit 124 which may be used to generate a digital clock aligned with certain CKV edges after the falling edge of FREF. The naming convention in this disclosure is as follows: non-uniform clock CKR is aligned with the first CKV falling edge after uniform reference clock FREF. CKRN is aligned with the N-th CKV falling edge after the falling edge of uniform reference clock FREF. The (average) frequency of non-uniform clock CKR 2 (and CKRN) is still the same as of uniform reference clock FREF, but the instantaneous period and start point of non-uniform clock CKR 2 (relative to uniform reference clock FREF) vary from cycle to cycle, depending on the DCO output signal CKV. This could result in significant error in the final DCO output phase ϕ V . The error due to the non-uniform clock CKR 2 is analyzed in FIG. 4 . Non-uniform clock CKR 2 introduces errors to the two paths 108 , 114 of the phase modulator in different ways. Firstly, for the DM-path 108 , the source of error lies in a variable accumulation time T acc [n] of the n-th cycle of non-uniform clock CKR 2 , during which the DCO 110 accumulates a constant frequency offset Δf M [n] to the desired phase change Δϕ M [n] of the n-th cycle of non-uniform clock CKR 2 . Under the uniform clock assumption, T acc [n]=T REF , the corresponding frequency change of the DCO 110 should be Δ ⁢ f M [ n ] = Δϕ M [ n ] T R ⁢ E ⁢ F . Equation ⁢ 4 This means that the desired phase change of Δϕ M [n] on oy can only be achieved if Δf M [n] is fed to the DCO 110 for exactly the duration of T REF . However, in general this is not the case as Δf M is updated with non-uniform clock CKR 2 in the circuit of FIG. 3 . Considering a time offset between non-uniform clock CKR 2 and Δf M update, D ana (propagation delay of control circuit and DCO response time), is almost a constant, T acc is equal to the period of non-uniform clock CKR 2 , which may vary from cycle to cycle. During the n-th Δf M update cycle, from t DCO [n] to t DCO [n+1], the phase modulation (PM) accumulated on the DCO 110 may be expressed as Δ ⁢ ϕ V ′ [ n ] = Δ ⁢ f M [ n ] · T a ⁢ c ⁢ c [ n ] . Equation ⁢ 5 An error relative to the desired phase Δϕ M [n] may be expressed as Δ ⁢ ϕ V , e ′ [ n ] = Δ ⁢ ϕ V ′ [ n ] - Δ ⁢ ϕ M [ n ] . Equation ⁢ 6 With Equation 4 and Equation 5, Equation 6 can be re-written as Δ ⁢ ϕ V , e ′ [ n ] = Δϕ M [ n ] T R ⁢ E ⁢ F · T a ⁢ c ⁢ c [ n ] - Δ ⁢ ϕ M [ n ] = - T R ⁢ E ⁢ F - T a ⁢ c ⁢ c [ n ] T R ⁢ E ⁢ F · Δϕ M [ n ] . Equation ⁢ 7 Δϕ′ V,e [n] is just the error of a single clock cycle. Considering the DCO 110 is a phase accumulator, all errors of all cycles may add up, i.e., at the moment t=t DCO [n], the overall accumulated PM may be expressed as ϕ V ′ ( t D ⁢ C ⁢ O [ n ] ) = ∑ i = 1 n - 1 { Δ ⁢ ϕ M [ i ] } + ∑ i = 1 n - 1 { Δ ⁢ ϕ V , e ′ [ i ] } = ϕ R ′ [ n - 1 ] + ∑ i = 1 n - 1 { Δ ⁢ ϕ V , e ′ [ i ] } . Equation ⁢ 8 The term Σ i=1 n−1 {Δϕ′ V,e [i]} indicates an error accumulation over time. Although the PLL limits such an error accumulation by correcting it via DCO tuning signal Δϕ T , this is usually a slow process compared to the data rate of the phase modulator 116 . Thus, the error will never be fully compensated by the PLL illustrated in FIG. 3 . Secondly, for the PP-path 114 , a source of error lies in the misalignment between uniform reference clock FREF falling edge (at t FREF [n]) and the update of Δf M (at t DCO [n]). In the ideal case ( FIG. 2 ), t DCO [n]=t FREF [n], which means that with phase prediction ϕ R [n−1] it's possible to precisely predict the time difference between FREF falling (or rising) edge and CKV falling (or rising) edge Δt S [n] (sampled at the falling edge of FREF), in order to detect the phase error in phase detector 118 Δ ⁢ t e [ n ] = ( 1 - ϕ R , f ⁢ r ⁢ a ⁢ c [ n - 1 ] ) · T C ⁢ K ⁢ V [ n ] - Δ ⁢ t S [ n ] . Equation ⁢ 9 However, in a realistic case, t DCO [n] and t FREF [n] may be significantly different. As indicated by Equation 8, the predicted phase ϕ′ R [n−1] is achieved around t DCO [n] thus, ϕ′ R [n−1] (or ϕ R [n−1]) is not an accurate prediction of ϕ′ V (t FREF [n]) (or ϕ V (t FREF [n])) any more. Therefore, a more accurate prediction of the of the DCO's output phase ϕ V,S [n] (or ϕ′ V,S in FIG. 4 ) is desired. Its fractional part ϕ V,S,frac , may then be used to detect the phase error Δ ⁢ t e [ n ] = ( 1 - ϕ V , S , frac [ n ] ) · T C ⁢ K ⁢ V [ n - 1 ] - Δ ⁢ t S [ n ] . Equation ⁢ 10 For example, a constant delay between the DCO's sampled output phase ϕ V,S and phase prediction ϕ R could be assumed to interpolate ϕ V,S . However, this assumption is not exact and can only achieve a sub-optimal estimation. The reasons are two-fold. Firstly, as indicated by Equation 8, phase prediction ϕ R [n−1] (for the beginning of n-th cycle) cannot be achieved exactly at t DCO [n] due to the accumulative error term. Hence, the delay between the DCO's sampled output phase ϕ V,S and phase prediction ϕ R is time-variant and (modulation) data dependent. Therefore, it may be important to consider the time-varying nature of the error to properly compensate it. Secondly, even if the error is so well compensated that ϕ R [n−1] is achieved exactly at t DCO [n], the delay between the DCO's sampled output phase ϕ V,S and phase prediction ϕ R (as well as the time offset between t FREF and t DCO ) is not a constant either. As shown in FIG. 4 , this delay may be defined as T update [ n ] = Δ ⁢ t S [ n ] + T C ⁢ K ⁢ V [ n - 1 ] + D a ⁢ n ⁢ a , Equation ⁢ 11 where Δt S [n] is the fractional timing error, T CKV [n−1] the CKV period, and D ana the analog delay due to DCO response time and propagation delay. The T CKV [n−1] and D ana terms are approximately constant, but the Δt S [n] may vary between 0 and T CKV [n−1]. Therefore, the constant delay assumption should be abandoned and a better estimation of the DCO's sampled output phase ϕ V,S should be achieved by correcting the DM-path error and considering the time-varying T update . The Non-Uniform Clock Compensation (NUCC) technique proposed in the present disclosure can suppress an error caused by these two mechanisms and thus improve the EVM of the phase modulator output. FIG. 5 illustrates a PLL circuit 500 for generating a modulated carrier signal in accordance with an embodiment of the present disclosure. PLL circuit 500 comprises DCO 110 configured to generate DCO output signal CKV as the modulated carrier signal. PLL circuit 500 comprises first input 102 configured to receive desired modulation phase ϕ M [n] or desired phase change Δϕ M [n] as a modulation signal at an n-th cycle of non-uniform clock CKR 2 derived from the DCO output signal CKV and a uniform reference clock FREF using snapshot circuit 124 . Optional differentiator circuitry 104 may be configured to differentiate the desired modulation phase ϕ M [n] to generate the desired phase change Δϕ M [n] at the n-th cycle of the non-uniform clock CKR 2 . PLL circuit 500 comprises the first DM path 108 for the desired phase change Δϕ M [n] to the DCO 110 to change its oscillation frequency based on the desired phase change Δϕ M [n]. PLL circuit 500 also comprises second input 112 configured to receive the frequency control word FCW at the n-th cycle of the non-uniform clock CKR 2 . That is, both inputs 102 , 112 of PLL circuit 500 are clocked or updated at non-uniform clock rate of non-uniform clock CKR 2 . PLL circuit 500 also comprises the second PP path 114 for the desired phase change Δϕ M [n] to frequency control word accumulator 116 which is configured to accumulate the desired phase change Δϕ M [n] and the FCW to generate a first phase prediction ϕ R [n] of the DCO's output phase at the beginning of the (n+1)-th cycle of the non-uniform clock CKR 2 . In contrast to the previous examples, PLL circuit 500 comprises a non-uniform clock compensation (NUCC) circuit 510 which is configured to recursively calculate a non-uniform clock compensation value ϕ DMC [n] 512 for the desired phase change Δϕ M [n] on the first path 108 and/or a second phase prediction ϕ S [n] 514 of the DCO's output phase at the (n+1)-th cycle of the uniform reference clock FREF. This recursive calculation may be done using the desired phase change Δϕ M [n], the frequency control word FCW, and the first phase prediction ϕ R [n] of frequency control word accumulator 116 . PLL circuit 500 comprises phase detector 518 configured to estimate an error Δt e [n] between (fractional part of) second phase prediction ϕ S [n] and an instantaneous time offset Δt S between the uniform reference clock FREF and the DCO output signal CKV (or CKR) and to generate a control signal for the DCO 110 based on the estimated error. The error Δt e [n] may be determined according to Δt e =(1−ϕ S,frac )·T CKV −Δt S . Thus, if ϕ S,frac =1, the error would be Δt e =−Δt S . If ϕ S,frac =0, the error would be Δt e =T CKV −Δt S . The error Δt e [n] may be fed to time-to-digital converter (TDC) 126 . Output of TDC 126 may be multiplied by a constant 1/K TDC and added to an integer part of second phase prediction Φ S,int [n] (which may be zero in locked state). A sampled phase ϕ V of the DCO output may be subtracted and the result may then be fed to loop filter 128 to obtain PLL control signal ΔΦ T [n] for DCO 110 at the n-th cycle of non-uniform clock CKR 2 . As mentioned above, the non-uniform digital clock CKR 2 may introduce errors on the phase modulator, mainly by diverging from the expected DCO phase accumulation duration T acc ≠T REF and the time offset between FREF falling edge and CKR 2 rising edge, i.e., when ϕ V reaches ϕ V,S and ϕ R . The basic idea of the proposed NUCC is that these variable parts can be well predicted. Thus, the relevant errors can be compensated. A key point of the PLL-based phase modulator with NUCC circuit 510 shown in FIG. 5 is the introduction of second phase prediction ϕ S in order to estimate the DCO's output phase ϕ V,S . ϕ S comprises a fractional part ϕ S,frac and an integer part ϕ S,int , same as the convention of first phase prediction ϕ R in FIG. 1 . Here, second phase prediction ϕ S [n−1] at the (n−1)-th cycle of non-uniform clock CKR 2 predicts ϕ V,S [n] at the (beginning of) n-th cycle of uniform reference clock FREF in FIG. 4 And with ϕ S,frac , Δt S , and consequently the variation of T update and T acc ( FIG. 4 ) can also be well estimated. Then, on the DM-path 108 , the estimated T acc may help to estimate the PM error ϕ′ V,e which may be compensated by non-uniform clock compensation value ϕ DMC . Because of the error compensation on the DM-path 108 , the delay between first phase prediction ϕ R and DCO's output phase ϕ V,S can be predicted by T update . Then with T update of the previous cycle, the proposed NUCC may give a good estimation of ϕ S,frac for the next cycle of non-uniform clock CKR 2 . The explanation above indicates that the compensation of the two paths 108 , 114 may rely on each other: the DM-path compensation may rely on an accurate prediction of Φ S or ϕ S,frac of the PP-path 114 to estimate T acc and the PP-path compensation may rely on the DM-path compensation to minimize the DCO's PM error ϕ′ V,e , thus, predicting ϕ S,frac more accurately. Therefore, these two compensations may be applied simultaneously to be more effective. However, the NUCC algorithm could also just include of one of the two compensation schemes, at the cost of having a larger residual error, which the PLL needs to compensate. Direct-Modulation-Path Compensation Firstly, consider the compensation on the DM-path 108 . Equation 7 has already estimated the error to the desired phase Δϕ M [n] accumulated in the DCO 110 , so the proposed NUCC just needs to compensate it in the next cycle of non-uniform clock CKR 2 with a non-uniform clock compensation value according to ϕ D ⁢ M ⁢ C [ n ] = - Δ ⁢ ϕ V , e ′ [ n - 1 ] = T R ⁢ E ⁢ F - T a ⁢ c ⁢ c [ n - 1 ] T R ⁢ E ⁢ F · Δϕ M [ n - 1 ] . Equation ⁢ 12 Thus, the NUCC circuit 510 may be configured to calculate the non-uniform clock compensation value ϕ DMC [n] for the n-th non-uniform clock CKR 2 cycle based on the desired phase change Δϕ M [n−1] of the (n−1)-th non-uniform clock CKR 2 cycle, a non-uniform period T acc [n−1] of the (n−1)-th non-uniform clock CKR 2 cycle, and a constant period T REF of the uniform reference clock. While the non-uniform period T acc [n−1] could be measured in some embodiments, the expression T R ⁢ E ⁢ F - T a ⁢ c ⁢ c [ n - 1 ] T R ⁢ E ⁢ F and hence ϕ DMC [n] may also be estimated using the second phase prediction ϕ S (or the fractional part ϕ S,frac thereof). Non-uniform clock compensation value ϕ DMC [n] may be added to the desired phase change Δϕ M [n] and thus impose an extra frequency change Δf DMC [n] on top of the original Δf M [n] to achieve the phase change of Δϕ M in each CKR 2 cycle. The extra frequency change Δf DMC [n] changes the slope of ϕ′ V (t) during the n-th cycle of non-uniform clock CKR 2 , as can be seen when comparing line 604 to original line 602 in FIG. 6 . Then, at the end of the n-th cycle (t DCO [n+1]), in addition to Δϕ M [n], an extra phase change close to ϕ DMC [n] has been integrated on the DCO output. Consequently, the error due to the previous cycle (Δϕ′ V,e [n−1]) is compensated. In terms of realizing this compensation technique, it may be needed to estimate the coefficient C D ⁢ M [ n ] = T F ⁢ R ⁢ E ⁢ F - T a ⁢ c ⁢ c [ n ] T F ⁢ R ⁢ E ⁢ F , Equation ⁢ 13 and Equation 12 consequently may be written as ϕ D ⁢ M ⁢ C [ n ] = C D ⁢ M [ n - 1 ] · Δϕ M [ n - 1 ] . Equation ⁢ 14 C DM [n] Estimation According to FIG. 6 , the instantaneous period T acc [n] of the n-th non-uniform clock CKR 2 cycle is T acc [n]=T REF −T update [n]+T update [n+1]. Considering Equation 11 and Δt S [n]=(1−ϕ S,frac [n−1])·T CKV [n−1], C DM [n] may be expressed as C D ⁢ M [ n ] = ( 2 - ϕ S , frac [ n - 1 ] ) ⁢ T C ⁢ K ⁢ V [ n - 1 ] T F ⁢ R ⁢ E ⁢ F - ( 2 - ϕ S , frac [ n ] ) ⁢ T C ⁢ K ⁢ V [ n ] T F ⁢ R ⁢ E ⁢ F . Equation ⁢ 15 Then according to the frequencies defined in the DCO model of FIG. 5 , T C ⁢ K ⁢ V [ n ] T F ⁢ R ⁢ E ⁢ F = f R ⁢ E ⁢ F f V [ n ] = f R ⁢ E ⁢ F f C + Δ ⁢ f M ⁢ W ⁢ C [ n ] = f R ⁢ E ⁢ F FCW · f R ⁢ E ⁢ F + Δ ⁢ ϕ M ⁢ W ⁢ C [ n ] · f R ⁢ E ⁢ F = 1 FCW + Δ ⁢ ϕ M ⁢ W ⁢ C [ n ] = 1 FCW + Δ ⁢ ϕ M [ n ] + ϕ D ⁢ M ⁢ C [ n ] . Equation ⁢ 16 Therefore, the coefficient C DM [n] for estimating the clock compensation value ϕ DMC [n] can be estimated as C D ⁢ M [ n ] = 2 - ϕ S , frac [ n - 1 ] F ⁢ C ⁢ W + Δ ⁢ ϕ M [ n - 1 ] + ϕ D ⁢ M ⁢ C [ n - 1 ] - 2 - ϕ S , frac [ n ] F ⁢ C ⁢ W + Δ ⁢ ϕ M [ n ] + ϕ D ⁢ M ⁢ C [ n ] . Equation ⁢ 17 Thus, the NUCC circuit 510 may be configured to calculate the coefficient C DM [n] based on: the frequency control word FCW, the (fractional part of) the second phase prediction ϕ S,frac of the (n−1)-th and the n-th cycle of the non-uniform clock CKR 2 , the desired phase change Δϕ M of the (n−1)-th and the n-th cycle of the non-uniform clock CKR 2 , and the non-uniform clock compensation value ϕ DMC of the (n−1)-th and the n-th cycle of the non-uniform clock CKR 2 . However, this might be too complex to implement in real applications when considering the comparably relaxed EVM requirements in current applications. If the condition FCW»Δϕ M »ϕ DMC is satisfied (which typically is the case), Equation 17 can be simplified to C D ⁢ M [ n ] ≈ ϕ S , frac [ n ] - ϕ S , frac [ n - 1 ] F ⁢ C ⁢ W . Equation ⁢ 18 Thus, the NUCC circuit 510 may be configured to calculate the coefficient C DM [n] based on: the frequency control word FCW, and the (fractional part of) the second phase prediction ϕ S,frac of the (n)-th and the (n−1)-th cycle of the non-uniform clock CKR 2 . Together with Equation 14, the NUCC circuit 510 may be configured to calculate the non-uniform clock compensation value ϕ DMC [n] for the n-th non-uniform clock CKR 2 cycle based on: the frequency control word FCW, the (fractional part of) the second phase prediction ϕ S,frac of the (n−2)-th and the (n−1)-th cycle of the non-uniform clock CKR 2 ; and the desired phase change Δϕ M of the (n−1)-th cycle of the non-uniform clock CKR 2 . The non-uniform clock compensation circuit may hence be configured to calculate the non-uniform clock compensation value ϕ DMC [n] for the n-th non-uniform clock CKR 2 cycle based on the second phase prediction ϕ S [n−2] of the DCO's output phase for the n−1-th uniform reference clock FREF cycle, the second phase prediction ϕ S [n−1] of the DCO's output phase for the n-th uniform reference clock FREF cycle, the frequency control word FCW, and the desired phase change Δϕ M [n−1] of the (n−1)-th non-uniform clock CKR 2 cycle. Such an approximation only results in a residual phase error at the DCO output in the order 1 FCW 2 . Extended Direct-Modulation-Path Compensation One may doubt Equation 12 is not accurate enough because there is also a tiny second order residual error due to the time-variance of T acc [n]. The expression Δ ⁢ f DMC [ n ] = ϕ DMC [ n ] T REF indicates that the exact phase compensation ϕ DMC [n] can only be achieved if T acc [n]=T REF . Due to the variation of T acc [n] there is a residual error, like the DM-path error Δ′ V,e . This error has the same form as Equation 7 ϕ DMC , e [ n ] = - T REF - T acc [ n ] T REF · Δϕ DMC [ n ] . Equation ⁢ 19 If this residual error shall be further compensated in the next cycle, a recursive extension of Equation 12 may be advantageous, for an even better estimation of ϕ DMC [n] Equation ⁢ 20 ϕ DMC [ n ] = - Δ ⁢ ϕ V , e ′ [ n - 1 ] - ϕ DMC , e [ n - 1 ] = T REF - T acc [ n - 1 ] T REF · ( Δϕ M [ n - 1 ] + ϕ DMC [ n - 1 ] ) = C DM [ n - 1 ] · ( Δϕ M [ n - 1 ] + ϕ DMC [ n - 1 ] ) . The NUCC circuit 510 may hence be configured to calculate the non-uniform clock compensation value ϕ DMC [n] for the n-th non-uniform clock CKR 2 cycle based on: the frequency control word FCW, the (fractional part of) the second phase prediction ϕ S,frac of the (n−2)-th and the (n−1)-th cycle of the non-uniform clock CKR 2 , the desired phase change Δϕ M of the (n−1)-th cycle of the non-uniform clock CKR 2 , and the non-uniform clock compensation value ϕ DMC [n−1] of the (n−1)-th non-uniform clock CKR 2 cycle. Then, depending on the required accuracy, either Equation 17 or Equation 18 can be used to estimate C DM [n]. One point to emphasis while using Equation 18 is that the assumption FCW»Δϕ M »ϕ DMC is used for this approximation. This assumption also indicates that the recursive term ϕ DMC [n−1] can be ignored here. Consequently, Equation 20 will fall back to the same form of Equation 14, which is ϕ DMC [ n ] = ϕ S , frac [ n - 1 ] - ϕ S , frac [ n - 2 ] FCW · Δϕ M [ n - 1 ] . Equation ⁢ 21 Phase-Prediction-Path Compensation Furthermore, consider the PP-path compensation. The target of PP-path compensation is to give an accurate estimation of the ϕ S [n], which predicts ϕ V (t) sampled at FREF falling edge, when t=t FREF [n+1]. ϕ S [n] may be estimated by interpolating ϕ R [n] which should ideally be reached around t DCO [n+1]. However, the uncompensated DM-path error blurs the position of ϕ R [n], thus making it difficult to interpolate ϕ S [n]. Luckily, this problem may be solved if we have the DM-path error compensation. With this compensation running in parallel, by the end of the n-th Δf MWC update cycle (when t=t DCP [n+1]), all the DM-path errors accumulated before this cycle have been eliminated. Thus Equation 8 can be modified to represent the ϕ′ V at this moment ϕ V ′ ( t DCO [ n + 1 ] ) = ϕ R ′ [ n ] + Δ ⁢ ϕ V , e ′ [ n ] . Equation ⁢ 22 The error term Δϕ′ V,e [n] shows up here because in general T acc [n]≠T REF . Therefore, if we virtually extend T acc [n] to T REF , which extrapolates ϕ′ V (t DCO [n+1]) to ϕ′ V (t DCO [n]+T REF ) with Δf MWC [n], the phase of ϕ′ R [n] can be achieved exactly (see line 606 and ϕ′ R [n] in FIG. 6 ). Considering the delay between the two moments when ϕ′ V reaches ϕ′ S [n] and ϕ′ R [n] is T REF −T acc,S [n]=T update [n], ϕ′ S [n] can be interpolated as ϕ S ′ [ n ] = ϕ R ′ [ n ] - T update [ n ] T REF · ( Δϕ M [ n ] + ϕ DMC [ n ] ) . Equation ⁢ 23 Taking the carrier into account, this can be written as ϕ S [ n ] = ϕ R [ n ] - T update [ n ] T REF · ( Δϕ M [ n ] + ϕ DMC [ n ] ) . Equation ⁢ 24 Thus, the NUCC circuit 510 may be configured to calculate the second phase prediction ϕ S [n] of the DCO's output phase for the (n+1)-th uniform reference clock (FREF) cycle based on the first phase prediction ϕ R [n] of the DCO's output phase at the (n+1)-th cycle of the non-uniform clock (CKR 2 ), the desired phase change Δϕ M [n] at the n-th cycle of a non-uniform clock CKR 2 , the non-uniform clock compensation value ϕ DMC [n], a period T REF of the uniform reference clock, and a non-uniform delay T update [n] between a (falling or rising) edge of the n-th uniform reference clock FREF cycle and a (rising or falling) edge of the (n+1)-th non-uniform clock CKR 2 cycle plus an essentially constant analog circuit delay D ana . To implement this estimation, the coefficient T update [ n ] T REF may be estimated. Following the similar scenario as the C DM analysis, T update [ n ] T REF may be estimated according to T update [ n ] T REF = 2 - ϕ S , frac [ n - 1 ] FCW + Δϕ M [ n - 1 ] + ϕ DMC [ n - 1 ] + D ana T REF . Equation ⁢ 25 The NUCC circuit 510 may hence be configured to estimate T update [ n ] T REF based on: the second phase prediction ϕ S,frac [n−1] of the DCO's output phase at the beginning of the n-th cycle of the uniform reference clock FREF, the desired phase change Δϕ M of the (n−1)-th cycle of the non-uniform clock CKR 2 ; the non-uniform clock compensation value ϕ DMC [n−1] of the (n−1)-th non-uniform clock CKR 2 cycle, the FCW, and and some essentially constant expression D ana T REF . Substituting Equation 25 into Equation 24 yields Equation ⁢ 26 ϕ S [ n ] = ϕ R [ n ] - ( 2 - ϕ S , frac [ n - 1 ] FCW + Δϕ M [ n - 1 ] + ϕ DMC [ n - 1 ] + D ana T REF ) · ( Δϕ M [ n ] + ϕ DMC [ n ] ) . The constant term D ana T REF can be estimated and the remaining variables are well-known from previous explanation. D ana denotes a constant delay between FREF and CKR 2 . Denote an estimate of D ana on chip as D est . If D est is larger or smaller, the ϕ′ S [n] will be shifted less or more towards the direction of Δϕ M [n]. This deviation can be detected by TDC 126 , whose output is D TDC [n]. Therefore, D est can be calibrated with a Least Mean Square (LMS) algorithm correlating Δϕ M [n] and D TDC [n]. Due to the integration effect of oscillator, actually correlated Δϕ M [n−1]·(1−z −1 ) and D TDC [n]. So D est may be D est =Σμ·(Δϕ M [n−1]−Δϕ M [n−2])·D est . With Equation 26, the second phase prediction ϕ S can be estimated accurately. But the hardware cost may be high due to its complexity. If the condition FCW»Δϕ M »ϕ DMC is satisfied (which typically is the case for a realistic design), Equation 26 can be approximated as ϕ S [ n ] ≈ ϕ R [ n ] - [ ( 2 FCW + D ana T REF ) - ϕ S , frac [ n - 1 ] FCW ] · ( Δϕ M [ n ] ) . Equation ⁢ 27 The constant term 2 FCW + D ana T REF here can still be estimated with the method described above. Comparing In Equation 27 to Equation 26, the error of the approximation is in the order of 1 FCW 2 . Since both Equation 26 and Equation 27 have recursive form, one might worry about the value of ϕ S,frac [n- 1 ] to be used for ϕ S [n] when n=0. Since a PLL-based phase modulator needs to lock the PLL before applying phase modulation, both Δϕ M [n] and ϕ DMC [n] may initially be set to 0. Then both these two equations degenerate to ϕ S [n]=ϕ R [n]. In other words, the initial value for ϕ S [0] may be ϕ S,frac [−1]=ϕ R,frac [−1]. If only Direct-Modulation-Path Compensation (without Phase-Prediction-Path Compensation) is used then the estimation of ϕ DMC may use ϕ R,frac instead of ϕ S,frac , i.e. ϕ DMC [ n ] = ϕ R , frac [ n - 2 ] - ϕ R , frac [ n - 1 ] FCW · Δϕ M [ n - 1 ] . If only Phase-Prediction-Path Compensation (without Direct-Modulation-Path Compensation) is used in the system, one may set ϕ DMC =0 in the corresponding equation, i.e. ϕ S [ n ] = ϕ R [ n ] - T update [ n ] T REF · ( Δϕ M [ n ] + 0 ) The proposed NUCC scheme can significantly improve the EVM of PLL-based phase modulator if it is not dominated by other analogue impairment, e.g., DCO nonlinearity. Unfortunately, DCO intrinsic nonlinearity, due to the inverse square root relationship between its oscillation frequency and variable capacitance, may have more significant influence on the EVM than the non-uniform clock. Therefore, the NUCC may work with some optional DCO nonlinearity digital pre-distortion (DPD) blocks shown in FIGS. 7 and 8 . FIG. 7 shows a system with both the NUCC and the DCO DPD. There are two possible positions to add DPD block 710 . DPD block 710 - 1 of option-1 is configured to pre-distort the phase signal Δϕ MWC in the DM-path after the DM-path compensation. This may be the most accurate way but may also introduce additional latency on ϕ DMC . Option 2 (DPD block 710 - 2 ) does not include ϕ DMC in the DCO DPD, thus, introducing less latency to ϕ DMC at the cost of accuracy. However, this is typically a favourable trade-off, since ϕ DMC needs to arrive at the DCO within one clock cycle and longer latency due to the DPD may degrade the compensation performance. Furthermore, ϕ DMC is relatively small. Excluding it from the DPD typically introduces negligible errors. FIG. 8 shows a further option for the DCO DPD. Unlike options 1 and 2, option 3 (DPD block 710 - 3 ) may have the advantage that it does not increase the loop latency since its output is fed into the DCO 110 in parallel with the signal path. Furthermore, it can be seen that the compensation of option 3 may be fed via a separate DCO bank, but it could also be added back to the modulation path or to the tracking path and feed it into the DCO via the respective DCO control bank. Also, all three DPD options now also have an input from the TDC output. This allows each DPD option to also “track” and compensate dynamic effects (e.g., transient effects of switching operations) and to compensate, not only static ones as before. In the system, each option could individually take on the full DPD, or one could have multiple options to share certain tasks. Embodiments of the present disclosure may improve the EVM to a level lower than approximately 20 ⁢ log 10 ( 1 FCW ) = 20 ⁢ log 10 ( f REF f c ) = 20 ⁢ log 10 ( f REF N · f c , ext ) , N being the frequency divider between an internal DCO rate f c and the desired external carrier frequency f c,ext . Because the variation of T acc and T update are within one T CKV cycle, which is around 1 FCW of T REF . Therefore, if these variations are not compensated, a distortion level around 20 ⁢ log 10 ( 1 FCW ) below the signal may always be present. For f c,ext =1 GHZ, f REF =40 MHz and N=4, embodiments of the present disclosure may have a residual error of <40 dB. This is of interest for modulation orders ≥1 k QAM, especially if some implementation margin shall be considered (e.g., all components should provide an error floor below the requirements, since all system errors will be summed-up for the total error observed at the output). 802.11ac (Wi-Fi 5) supports up to 256-QAM where 8 bits of data are encoded per symbol EVM <−30 dB 802.11ax (Wi-Fi 6) supports up to 1024-QAM, where 10 bits of data are encoded per symbol EVM <−35 dB 802.11be introduces 4096-QAM modulation with 12 bits of data per symbol EVM <−38 dB Note that the present technology can also be configured as described below. Example 1 is a PLL circuit for generating a modulated carrier signal. The PLL circuit comprises a controlled oscillator configured to generate a controlled oscillator output signal CKV as the modulated carrier signal. A first input is configured to receive a desired phase change as a modulation signal at an n-th cycle of a non-uniform clock derived from the controlled oscillator output signal and a uniform reference clock. A first path for the desired phase change leads to the controlled oscillator to change its oscillation frequency based on the desired phase change. A second input configured to receive a frequency control word (FCW) at the n-th cycle of the non-uniform clock. The frequency control word (FCW) is a ratio of a desired carrier frequency of the carrier signal and a frequency of the uniform reference clock. A second path for the desired phase change leads to a frequency control word accumulator configured to accumulate the desired phase change and the FCW to generate a first phase prediction of the controlled oscillator's output phase at the (n+1)-th cycle of the non-uniform clock. A non-uniform clock compensation circuit is configured to recursively calculate, using the desired phase change, the frequency control word FCW and the first phase prediction, a non-uniform clock compensation value for the desired phase change on the first path and/or a second phase prediction of the controlled oscillator's output phase at the (n+1)-th cycle of the uniform reference clock. A phase detector is configured to estimate an error between the second phase prediction and an instantaneous time offset between the uniform reference clock and the controlled oscillator output signal and to generate a control signal for the controlled oscillator based on the estimated error. In Example 2, the controlled oscillator output signal and the uniform reference clock of Example 1 have different respective frequencies and are asynchronous signals. In Example 3, the non-uniform clock and the uniform reference clock of Examples 1 and 2 have equal respective average frequencies and instantaneous periods and start points of the non-uniform clock relative to the uniform reference clock vary from cycle to cycle. In Example 4, the non-uniform clock of any one of Examples 1 to 3 is phase aligned with the controlled oscillator's output signal. In Example 5, the first input of any one of Examples 1 to 4 is configured to receive a desired modulation phase for the modulated carrier signal at the n-th cycle of the non-uniform clock. The PLL circuit further comprises a differentiator circuit configured to differentiate the desired modulation phase to generate the desired phase change at the n-th cycle of the non-uniform clock. In Example 6, the phase detector of any one of Examples 1 to 5 is configured to estimate the error between a predicted time difference between two nearest falling/rising edges of the uniform reference clock and the controlled oscillator's output signal and a measured time difference between two nearest falling/rising edges of the uniform reference clock and the DCO output signal. In Example 7, the non-uniform clock compensation circuit of any one of Examples 1 to 6 is configured to calculate the non-uniform clock compensation value for the n-th non-uniform clock cycle based on the desired phase change of the (n−1)-th non-uniform clock cycle, an instantaneous period of the n-th non-uniform clock cycle, and a period of the uniform reference clock. In Example 8, the non-uniform clock compensation circuit of any one of Examples 1 to 7 is configured to calculate the non-uniform clock compensation value (ϕ DMC [n]) for the n-th non-uniform clock cycle based on the second phase prediction (ϕ S [n−2]) of the controlled oscillator's output phase for the (n−1)-th uniform reference clock (FREF) cycle, the second phase prediction (ϕ S [n−1]) of the DCO's output phase for the n-th uniform reference clock (FREF) cycle, the frequency control word FCW; and the desired phase change (Δϕ M [n−1]) of the (n−1)-th non-uniform clock (CKR 2 ) cycle. In Example 9, the non-uniform clock compensation circuit of any one of Examples 1 to 8 is configured to calculate the non-uniform clock compensation value (ϕ DMC [n]) for the n-th non-uniform clock (CKR 2 ) cycle based on ϕ DMC [ n ] = ϕ S , frac [ n - 1 ] - ϕ S , frac [ n - 2 ] FCW · Δϕ M [ n - 1 ] , wherein ϕ S,frac [·] denotes a fractional part of second phase prediction ϕ S [□] used to track the phase of the modulated carrier signal when the PLL circuit operates in a locked state. In Example 10, the non-uniform clock compensation circuit of any one of Examples 1 to 9 is configured to calculate the second phase prediction (ϕ S [n]) of the DCO's output phase for the (n+1)-th uniform reference clock (FREF) cycle based on the first phase prediction (ϕ R [n]), the desired phase change (Δϕ M [n]), the non-uniform clock compensation value (ϕ DMC [n]), a period (T REF ) of the uniform reference clock, and a non-uniform delay (T update [n]) between an edge of the n-th uniform reference clock (FREF) cycle and an edge of the (n+1)-th non-uniform clock (CKR 2 ) cycle plus a constant analog circuit delay (D ana ). In Example 11, the non-uniform clock compensation circuit of Example 10 is configured to calculate the second phase prediction (ϕ S [n]) of the DCO's output phase for the (n+1)-th uniform reference clock (FREF) cycle based on ϕ S [ n ] = ϕ R [ n ] - T update [ n ] T REF · ( Δϕ M [ n ] + ϕ DMC [ n ] ) . In Example 12, the non-uniform clock compensation circuit of Example 10 or 11 is configured to calculate the second phase prediction (ϕ S [n]) of the DCO's output phase for the (n+1)-th uniform reference clock (FREF) cycle based on ϕ S [ n ] ≈ ϕ R [ n ] - [ ( 2 FCW + D ana T REF ) - ϕ S , frac [ n - 1 ] FCW ] · ( Δϕ M [ n ] ) , wherein ϕ S,frac [·] denotes a fractional part of ϕ S [·] used to track the phase of the modulated carrier signal when the PLL circuit operates in a locked state. Example 13 is a polar transmitter comprising a PLL circuit of any one of Examples 1 to 12. Example 14 is a method for generating a modulated carrier signal. The method includes providing a controlled oscillator configured to generate a controlled oscillator output signal (CKV) as the modulated carrier signal. The method includes receiving a desired phase change (Δϕ M [n]) as a modulation signal at an n-th cycle of a non-uniform clock (CKR 2 ) derived from the controlled oscillator output signal (CKV) and a uniform reference clock (FREF) of a PLL circuit. The method includes providing the desired phase change (Δϕ M [n]) to the controlled oscillator to change its oscillation frequency based on the desired phase change (Δϕ M [n]). The method includes receiving a frequency control word (FCW) at the n-th cycle of the non-uniform clock (CKR 2 ), the frequency control word (FCW) being a ratio of a desired carrier frequency (f c ) of the carrier signal and a frequency (f REF ) of the uniform reference clock. The method includes providing the desired phase change (Δϕ M ) to a frequency control word accumulator configured to accumulate the desired phase change (Δϕ M [n]) and the FCW to generate a first phase prediction (Δ R [n]) of the controlled oscillator's output phase at the (n+1)-th cycle of the non-uniform clock (CKR 2 ). The method includes recursively calculating, using the desired phase change (Δϕ M [n]), the frequency control word FCW, and the first phase prediction (Δ R [n]), a non-uniform clock compensation value (ϕ DMC [n]) for the desired phase change Δϕ M [n] on the first path and/or a second phase prediction (ϕ S [n]) of the controlled oscillator's output phase at the (n+1)-th cycle of the uniform reference clock (FREF). The method includes estimating an error between the second phase prediction (ϕ S [n]) and an instantaneous time offset (Δt S ) between the uniform reference clock (FREF) and the controlled oscillator output signal (CKV) and generating a control signal for the controlled oscillator based on the estimated error. The aspects and features described in relation to a particular one of the previous examples may also be combined with one or more of the further examples to replace an identical or similar feature of that further example or to additionally introduce the features into the further example. Examples may further be or relate to a (computer) program including a program code to execute one or more of the above methods when the program is executed on a computer, processor or other programmable hardware component. Thus, steps, operations or processes of different ones of the methods described above may also be executed by programmed computers, processors or other programmable hardware components. Examples may also cover program storage devices, such as digital data storage media, which are machine-, processor- or computer-readable and encode and/or contain machine-executable, processor-executable or computer-executable programs and instructions. Program storage devices may include or be digital storage devices, magnetic storage media such as magnetic disks and magnetic tapes, hard disk drives, or optically readable digital data storage media, for example. Other examples may also include computers, processors, control units, (field) programmable logic arrays ((F)PLAs), (field) programmable gate arrays ((F)PGAs), graphics processor units (GPU), application-specific integrated circuits (ASICs), integrated circuits (ICs) or system-on-a-chip (SoCs) systems programmed to execute the steps of the methods described above. It is further understood that the disclosure of several steps, processes, operations or functions disclosed in the description or claims shall not be construed to imply that these operations are necessarily dependent on the order described, unless explicitly stated in the individual case or necessary for technical reasons. Therefore, the previous description does not limit the execution of several steps or functions to a certain order. Furthermore, in further examples, a single step, function, process or operation may include and/or be broken up into several sub-steps,-functions, -processes or -operations. If some aspects have been described in relation to a device or system, these aspects should also be understood as a description of the corresponding method. For example, a block, device or functional aspect of the device or system may correspond to a feature, such as a method step, of the corresponding method. Accordingly, aspects described in relation to a method shall also be understood as a description of a corresponding block, a corresponding element, a property or a functional feature of a corresponding device or a corresponding system. The following claims are hereby incorporated in the detailed description, wherein each claim may stand on its own as a separate example. It should also be noted that although in the claims a dependent claim refers to a particular combination with one or more other claims, other examples may also include a combination of the dependent claim with the subject matter of any other dependent or independent claim. Such combinations are hereby explicitly proposed, unless it is stated in the individual case that a particular combination is not intended. Furthermore, features of a claim should also be included for any other independent claim, even if that claim is not directly defined as dependent on that other independent claim. Definition of Symbols Δt S the instantaneous time offset between the falling edges FREF and subsequent CKV T REF the period of FREF, which is the frequency reference for PLL T CKV the period of CKV, which is the output of DCO T acc the duration between Δf M update. The DCO keeps the frequency change of for the duration of T acc to accumulate the desired phase change T acc,S the duration between Δf M update and FREF falling. During this period, the phase of ϕ′ V changes from ϕ′ R,A to ϕ′ S,A T update the time offset between FREF falling and Δf M update D ana the delay between CKR 2 and Δf M or Δf MWC settling (Caution: all the phases here have been normalized with 1 2 ⁢ π ) ϕ V output phase of DCO ϕ S prediction for the ϕ V sampled by FREF falling edge ϕ R prediction for the ϕ V at the moment of Δf M update if no errors caused by the non-uniform digital clock ϕ′ V the ϕ V excluding the phase of carrier Δϕ′ V the phase change of ϕ′ v during T acc (the duration between Δf M update) ϕ′ s the ϕ S excluding the predicted phase of carrier ϕ′ R the ϕ R excluding the predicted phase of carrier ϕ DMC the phase of DM-Path Compensation ϕ DMC,e the error between the desired ϕ DMC and the achieved value ϕ M the desired phase for modulation Δϕ M the change of ϕ M during each FREF/CKR 2 cycles Δϕ T the output of loop filter to help DCO track f c Δϕ′ V,e the error between the desired phase Δϕ M and the accumulated phase on DCO Δf m the frequency change on DCO to achieve phase change of Δϕ M in each FREF/CKR 2 cycles Δf DMC the frequency change on DCO to compensate ϕ DMC Δf MWC the frequency change Δf M with compensation Δf DMC f V instantaneous frequency of DCO f C target carrier frequency of the phase modulator f CO intrinsic frequency of DCO around the f c . It should be corrected by the f T to reach f c Δf T the frequency change of DCO in response to Δϕ T f REF frequency of FREF